Triangle primes

[Andrew Plewe]

I plan to submit the following sequence to the OEIS. These are Triangle
primes, or primes which can be expressed as the sum of two triangle numbers:


13, 31, 43, 61, 73, 83, 97, 101, 127, 139, 151, 157, 163, 181, 191, 193,
199, 211, 227, 241, 263, 281, 307, 331, 353, 367, 373, 379, 409, 421, 433,
461, 463, 487, 499, 541, 571, 577, 587, 601, 619, 631, 659, 661, 673, 709,
727, 739, 751, 757, 769, 821, 823, 839, 853, 877, 883, 911, 919, 967, 991,
997, 1033, 1039, 1051


Would someone be so kind as to verify that the terms I have are correct? I
have checked these by hand by referencing a list of the first 1000 primes
from the Prime Pages website, a process which is prone to errors.


I find this sequence interesting for several reasons. First, more primes
than I'd originally expected show up in the list. Second, I derived this
list from a table of sums of triangle numbers. That table seems to have some
interesting properties. One of these is a simple prime finding/proving
method which may or may not (I don't have a proof yet) work for all odd
integers in the table. The method is essentially the following:


1.) Look for a Sophie Germain "pair" (i.e. n and 2n +/- 1). The product of
any Sophie Germain "pair" is also a triangle number. For my example I'll use
3779 * 1889

2.) Find another number which is prime. In this case I'll use 5197. Divide
that number plus or minus one by two, the dividend should be an even number.
5196 / 2 = 2598 will work. The product of this "pair" is also a triangle
number.

3.) Add the two triangle numbers. 3779 * 1889 + 5197 * 2598 = 20640337. This
is our prime "candidate".

4.) Perform a GCD with the candidate and the two triangle numbers. If GCD >
1, the number is composite. In this case, GCD = 1 for both triangle numbers.

5.) Find all of the "neighboring" triangle sums immediate around the
candidate value. This can be done by finding the triangle numbers immediate
before and after our "pairs",:


 3777 * 1889
 3779 * 1889
 3779 * 1890 and

 5195 * 2598
 5197 * 2598
 5197 * 2599

and adding together all possible combinations of the pairs (excluding adding
a pair to itself):

3777 * 1889 + 5195 * 2598 = 20631363
3777 * 1889 + 5195 * 2599 = 20635141 ... etc.

6.) Now, find the difference of each neighbor and the candidate value:

 20640337 - 20631363 = 8974
 20640337 - 20635141 = 5196 ... etc.

7.) Create a set of sums and differences of these values:

  8974 - 5196 = 3778
  8974 + 5196 = 14170 ... etc.

8.) GCD all values in the set of sums and differences with the candidate
value. If a gcd > 1 is found, the number is composite. if not, the number is
prime. In this case, the number is composite:

  8976 + 5196 = 14172, gcd(14172, 20640337) = 1181


Another interesting thing (to me, anyway) is that if there are not an
infinite number of Sophie Germain primes then at some point in time the
"density" of primes in the sum table of triangle numbers will drop
drastically. Anyway, I apologize for the length of this email if you're
bored to tears by this, hopefully some of you find this interesting. Thanks!

	-Andrew Plewe-